Name: reynolds number from buckingham pi
Uploaded: 2025-02-13T15:43:29.000Z
Duration: 1 min 9 s
Channel: JoVE

Betrachten Sie die Strömung durch ein Rohr, wobei die Schlüsselvariablen der Rohrdurchmesser D, die Flüssigkeitsgeschwindigkeit V, die Dichte ρ und die Viskosität μ sind.
Um das System mit dem Buckingham-Pi-Theorem zu analysieren, können diese Variablen zu dimensionslosen Produkten wie der Reynolds-Zahl kombiniert werden, die das Strömungsverhalten charakterisiert.
In diesem Fall wird das Buckingham-Pi-Theorem angewendet, indem zunächst die Anzahl der beteiligten Variablen und die Referenzdimensionen identifiziert werden, die Masse M, Länge L und Zeit T sind.
Nach dem Buckingham-Pi-Theorem ergibt das System mit vier Variablen und drei Referenzdimensionen (M, L, T) einen dimensionslosen Pi-Term, der die Reynolds-Zahl ist.
Hier ist die Reynolds-Zahl ein dimensionsloses Produkt, das die Trägheitskräfte mit den viskosen Kräften in der Flüssigkeit in Beziehung setzt, was zur Klassifizierung der Strömung als laminar oder turbulent beiträgt.
Das Buckingham-Pi-Theorem formuliert das System neu, indem es die Variablen als dimensionslose Terme ausdrückt, was es einfacher macht, die Strömung unter unterschiedlichen Bedingungen zu verallgemeinern und zu analysieren.

der buckingham pi-satz

The Buckingham Pi Theorem

Der Buckingham Pi-Satz bietet eine strukturierte Methode zur Vereinfachung von Problemen der Strömungsmechanik, indem komplexe Variablensysteme auf dimensionslose Terme reduziert werden.

<img alt="Equation 1" src="/files/ftp_upload/18068/18068_Equation_1.svg" style="text-align: center; height: 34px;">


Bei der Analyse des Durchflusses durch ein Rohr umfassen die Variablen, die das System definieren, den Rohrdurchmesser, die Geschwindigkeit des Fluids, seine Dichte und Viskosität. Diese können zu einem einzigen dimensionslosen Term kombiniert werden, der Reynolds-Zahl (Re), die ihre wesentliche Beziehung erfasst.

<img alt="Equation 2" src="/files/ftp_upload/18068/18068_Equation_2.svg" style="text-align: center; height: 38px;">


Der erste Schritt bei der Anwendung des Buckingham Pi-Satzes besteht darin, die grundlegenden Dimensionen jeder Variable zu identifizieren. In der Strömungsmechanik sind dies normalerweise Masse, Länge und Zeit. Hier stellt der Durchmesser die Länge dar, die Geschwindigkeit kombiniert Länge und Zeit, die Dichte umfasst Masse und Volumen und die Viskosität umfasst Masse, Länge und Zeit.

<img alt="Equation 3" src="/files/ftp_upload/18068/18068_Equation_3.svg" style="text-align: center; height: 140px;">


Der Satz legt nahe, dass die Anzahl der unabhängigen dimensionslosen Gruppen, bekannt als Pi-Terme, die Differenz zwischen der Anzahl der Variablen k und der Anzahl der Grunddimensionen r ist. Bei vier Variablen und drei Dimensionen wird nur ein dimensionsloser Term benötigt, was die Analyse vereinfacht.

<img alt="Equation 4" src="/files/ftp_upload/18068/18068_Equation_4.svg" style="text-align: center; height: 32px;">

Dieser einzelne Term, die Reynolds-Zahl, vergleicht Trägheitskräfte mit viskosen Kräften in den Fluiden. Die Ermittlung dieses Verhältnisses zeigt, ob die Strömung wahrscheinlich glatt und geordnet ist (laminare Strömung), oder chaotisch und unregelmäßig (turbulente Strömung). Niedrigere Werte weisen auf laminare Strömung hin, bei der viskose Kräfte dominieren und sich das Fluid in parallelen Schichten bewegt. Höhere Werte spiegeln turbulente Strömung wider, bei der Trägheitskräfte zu chaotischer Bewegung führen. Die Anwendung des Buckingham Pi-Satzes in diesem Zusammenhang reduziert komplexe Interaktionen auf einen einzigen Term, wodurch es einfacher wird, das Strömungsverhalten unter verschiedenen Bedingungen zu verallgemeinern.

Der Buckingham Pi-Satz

Betrachten Sie die Strömung durch ein Rohr, wobei die Schlüsselvariablen der Rohrdurchmesser D, die Flüssigkeitsgeschwindigkeit V, die Dichte ρ und die Viskosität μ sind.
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reynolds number from buckingham pi

The Buckingham Pi theorem turns fluid flow variables into dimensionless terms. In a pipe flow example, the key variables are the pipe diameter, fluid velocity, density, and viscosity. These variables can be combined into one dimensionless group, the Reynolds number, which captures their relationship.

Applying the theorem starts with identifying the fundamental dimensions of each variable. In fluid mechanics, these dimensions are mass, length, and time. The pipe diameter represents length, velocity combines length and time, density includes mass and volume, and viscosity includes mass, length, and time.

The theorem also shows how many independent dimensionless groups, or Pi terms, are needed. That number is found by subtracting the number of fundamental dimensions, r, from the number of variables, k. With four variables and three dimensions, only one dimensionless term is needed.

That single term is the Reynolds number, often written as Re. It compares inertial forces to viscous forces in the fluid. Low values point to laminar flow, where the fluid moves in smooth, parallel layers. Higher values point to turbulent flow, where motion becomes chaotic and irregular.

This makes the Buckingham Pi theorem useful for simplifying fluid dynamics problems. It reduces several linked variables to one term that can be used to compare flow behavior under different conditions. That simplification makes it easier to generalize how fluids move through a pipe.

Reynolds Number from Buckingham Pi

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Reynolds Number from Buckingham Pi

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